24 DR GUSTAVE PLARR ON THE 



Our next step will be to eliminate {yy') u from W. We take 



_ [v i ( -i)* +y n& u 2p 



This gives 



zi"=[2>2;-'i2>2: ? ]W", 



where, by reductions easily seen we have 



( - 1 )'J+r+m(2n - 2g)Jlk.^- s -^- 2 P(x') n -*-*-»« 



W" = , 



2»+'+»n0lI(w - g)IL(n -s-2g- 2/c)II(s + Jc)TlpU(k -p)UqU(k - q) ' 

 In the place of p, q we introduce the indices u, v, respectively by the relations 



g-\-p = U, g + q = v. 



The limits of u will be 



u=g and io=g + k 



v=g and v=g + k. 



As & does not depend on p, q we replace it by the index I by the relation 



g+k = l. 



The limits of I will be I = g for k = 0, and Z = r for & = r—g . The limits of u and v will 

 then be u = g and u — I, and the same for v. 

 We have now 



~ g 2 £ z,u 2 



IT <7 ? 3 



where 



W'" = 



g g g -> 



(-l)o+"+"U(2n-2g)Il(l-g)2- 2 <' 



2 n+ '+ 2l UgIl(n - g)Tl(n - s - 2l)U(s +l-g) U(u - g)U(l - u)U(v - g)U(l - v) 



§11. 



The quadruple sums 2 must now be submitted to interversions so as to bring the. 

 sums relative to u and v to the left of those relative to g, I. 



